i dont really understand the question. your picture seem contrary to your description.
your picture seems to suggest you can cut in any direction, but your description suggests all cuts have to be in the same direction.
i'll solve both.
for lines in any direction: http://mathworld.wolfram.com/CircleDivisionbyLines.html
1/2*(n^2 +n +2), n=500; 125251
5000000 = 1/2*(n^2 +n +2) n=3162
for cuts in the same direction:
each cut adds 1 new region.
for 500 cuts, that 501 rejoins
to get 5000000 peices 4999999 cuts are needed

heres an alternate version that i came up with,
place the digits 1-n where n is 4 such that no consecutive digit of any step value, starting at step value, repeats.
here's an example where n is 2.
0 1 0 2 1 0 2 1 2 0 1 ?
1 2 3 4 5 6 7 8 9 10 11 12
here starting from 1 and going a step of 1, there are no repeats. starting from 2 and going a step of 2, no repeats, and so on. However there is no way to get 12 without repeating.
your task is to find the max value for 4.